Paper · Apr 25, 2016

Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics

Tamás Darvas

math.DGarXiv:1604.07127v2
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Abstract

Let be a compact normal Kähler space, with Hodge metric . In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on , the space of Kähler potentials, whose completion is the finite energy space . Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting Kähler-Einstein metrics. We also discuss the analogous result for Kähler-Ricci solitons on Fano varieties.

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Submitted
Apr 25, 2016
Last revised
Oct 25, 2016 (v2)
Subjects
math.DG, math.CV
Journal
IMRN (2017), no. 22, 6752-6777

Comments: v2 Final version. J. Streets discovered that the approximation result of [EGZ2] has incomplete proof. As a result, the approximation result of [CGZ] has to be used, that applies only in the case of Hodge Kahler metrics. We modified the statements of Theorem 2.1 and Theorem 3.5 accordingly. The main applications regarding canonical Kahler metrics are unchanged

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability

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  6. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

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    Tamás Darvas

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.